Automatic Mathematical Information Retrieval to Perform Translations - - PowerPoint PPT Presentation

Automatic Mathematical Information Retrieval to Perform Translations up to Computer Algebra Systems

André Greiner-Petter August 13, 2018

University of Konstanz Germany @GreinerPetter 1/9

Motivation & Problems

Motivation - Formulae Presentations

DLMF 18.3

A Jacobi polynomial in different systems. Rendered Version: P (α,β)

n

(cos(aΘ)) Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple: JacobiP(n,alpha,beta,cos(a*Theta)) CAS Mathematica: JacobiP[n,\[Alpha],\[Beta],Cos[a \[CapitalTheta]]]

2/9

Motivation - Formulae Presentations

DLMF 18.3

A Jacobi polynomial in different systems. Rendered Version: P (α,β)

n

(cos(aΘ)) Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple: JacobiP(n,alpha,beta,cos(a*Theta)) CAS Mathematica: JacobiP[n,\[Alpha],\[Beta],Cos[a \[CapitalTheta]]]

2/9

Motivation - Formulae Presentations

DLMF 18.3

A Jacobi polynomial in different systems. Rendered Version: P (α,β)

n

(cos(aΘ)) Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple: JacobiP(n,alpha,beta,cos(a*Theta)) CAS Mathematica: JacobiP[n,\[Alpha],\[Beta],Cos[a \[CapitalTheta]]]

2/9

Motivation - Formulae Presentations

DLMF 18.3

A Jacobi polynomial in different systems. Rendered Version: P (α,β)

n

(cos(aΘ)) Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple: JacobiP(n,alpha,beta,cos(a*Theta)) CAS Mathematica: JacobiP[n,\[Alpha],\[Beta],Cos[a \[CapitalTheta]]]

2/9

Presentation To Computation with semantic information

Problems of Translations

DLMF 18.3

Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple: JacobiP(n, alpha, beta, cos(a*Theta)) Potential Problems:

• Differences in syntax
• Function is not implemented in one system,
• Function has multiple representations in one system,
• Differences in definitions.

3/9

Problems of Translations

DLMF 18.3

Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple: JacobiP(n, alpha, beta, cos(a*Theta)) Potential Problems:

• Differences in syntax
• Function is not implemented in one system,
• Function has multiple representations in one system,
• Differences in definitions.

3/9

Problems of Translations

DLMF 18.3

Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple: JacobiP($2, $0, $1, $3) Potential Problems:

• Differences in syntax

← solved by translation patterns

• Function is not implemented in one system,
• Function has multiple representations in one system,
• Differences in definitions.

3/9

Problems of Translations

DLMF 18.3

Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple: JacobiP(n, alpha, beta, cos(a*Theta)) Potential Problems:

• Differences in syntax

← solved by translation patterns

• Function is not implemented in one system,
• Function has multiple representations in one system,
• Differences in definitions.

3/9

Problems of Translations

DLMF 18.3

Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple:

DLMF 18.5.7

Potential Problems:

• Differences in syntax

← solved by translation patterns

• Function is not implemented in one system,
• translate equivalent presentations
• Function has multiple representations in one system,
• Differences in definitions.

n

• ℓ=0

(n + α + β + 1)ℓ(α + ℓ + 1)n−ℓ ℓ! (n − ℓ)!

x − 1

2

ℓ

3/9

Problems of Translations

DLMF 18.3

Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple: JacobiP or Jacobi or JacobiPoly Potential Problems:

• Differences in syntax

← solved by translation patterns

• Function is not implemented in one system,
• translate equivalent presentations
• Function has multiple representations in one system,
• Differences in definitions.

3/9

Problems of Translations

DLMF 18.3

Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple: JacobiP or Jacobi or JacobiPoly Potential Problems:

• Differences in syntax

← solved by translation patterns

• Function is not implemented in one system,
• translate equivalent presentations
• Function has multiple representations in one system,
• just pick a valid translation
• Differences in definitions.

3/9

Problems of Translations

DLMF 18.3

Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple: JacobiP(n, alpha, beta, cos(a*Theta)) Potential Problems:

• Differences in syntax

← solved by translation patterns

• Function is not implemented in one system,
• translate equivalent presentations
• Function has multiple representations in one system,
• just pick a valid translation
• Differences in definitions.

3/9

Problems of Translations

DLMF 18.3

Semantic L

AT

EX: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} CAS Maple: JacobiP(n, alpha, beta, cos(a*Theta)) Potential Problems:

• Differences in syntax

← solved by translation patterns

• Function is not implemented in one system,
• translate equivalent presentations
• Function has multiple representations in one system,
• just pick a valid translation
• Differences in definitions. ← wait... What?

3/9

Problems of Translations

DLMF 4.23.9 Maple Inv. Trig. Functions

Rendered Version Semantic L

AT

EX CAS Maple arccot(z) \acot@{z} arccot(z)

4/9

Problems of Translations

DLMF 4.23.9 Maple Inv. Trig. Functions

Rendered Version Semantic L

AT

EX CAS Maple arccot(z) \acot@{z} arccot(z) Maple

Figure 1: ℜ(arccot(z)) with branch cut at [−∞i, −i], [i, ∞i].

DLMF & Mathematica

Figure 2: ℜ(arccot(z)) with branch cut at [−i, i].

4/9

Problems of Translations

DLMF 4.23.9 Maple Inv. Trig. Functions

Rendered Version Semantic L

AT

EX CAS Maple arccot(z) \acot@{z} arccot(z) Maple

Figure 1: ℜ(arccot(z)) with branch cut at [−∞i, −i], [i, ∞i].

DLMF & Mathematica

Figure 2: ℜ(arccot(z)) with branch cut at [−i, i].

4/9

Problems of Translations

DLMF 4.23.9 Maple Inv. Trig. Functions

Rendered Version Semantic L

AT

EX CAS Maple arccot(z) \acot@{z} arctan(1/z) Maple

Figure 1: ℜ(arccot(z)) with branch cut at [−∞i, −i], [i, ∞i].

DLMF & Mathematica

Figure 2: ℜ(arccot(z)) with branch cut at [−i, i].

4/9

Presentation To Computation (P2C) without semantic information

Problems of Generic L

A

T EX

DLMF 18.3

Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantics: \JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}} Potential Problems:

• Is P a function, variable, constant?
• Is cos(aΘ) an argument of P or part of a multiplication?
• What are α, β, n, a, and Θ?

5/9

Problems of Generic L

A

T EX

DLMF 18.3

Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantics: Jacobi polynomial or Legendre function or Ferrers function or ... Potential Problems:

• Is P a function, variable, constant?
• Is cos(aΘ) an argument of P or part of a multiplication?
• What are α, β, n, a, and Θ?

5/9

Problems of Generic L

A

T EX

DLMF 18.3

Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantics: P(cos(aΘ)) vs P · (cos(aΘ)) Potential Problems:

• Is P a function, variable, constant?
• Is cos(aΘ) an argument of P or part of a multiplication?
• What are α, β, n, a, and Θ?

5/9

Problems of Generic L

A

T EX

DLMF 18.3

Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Semantics: Variable or 2nd Feigenbaum constant or ... Potential Problems:

• Is P a function, variable, constant?
• Is cos(aΘ) an argument of P or part of a multiplication?
• What are α, β, n, a, and Θ?

5/9

Human Approach

Rendered L

AT

EX: P (α,β)

n

(cos(aΘ))

6/9

Human Approach

Rendered L

AT

EX: P (α,β)

n

(cos(aΘ)) The Naive Approach How does a reader understands the mathematical formula?

• he knows the symbols and structure,
• knowledge-based pattern recognition
• it was previously introduced in the paper (e.g. in definitions,

the text or in other referenced publications),

• analyse the context from near to far
• he searching the formula in books or online
• dictionary-based pattern recognition

6/9

Human Approach

Rendered L

AT

EX: P (α,β)

n

(cos(aΘ)) The Naive Approach How does a reader understands the mathematical formula?

• he knows the symbols and structure,
• knowledge-based pattern recognition
• it was previously introduced in the paper (e.g. in definitions,

the text or in other referenced publications),

• analyse the context from near to far
• he searching the formula in books or online
• dictionary-based pattern recognition

6/9

Human Approach

Rendered L

AT

EX: P (α,β)

n

(cos(aΘ)) The Naive Approach How does a reader understands the mathematical formula?

• he knows the symbols and structure,
• knowledge-based pattern recognition
• it was previously introduced in the paper (e.g. in definitions,

the text or in other referenced publications),

• analyse the context from near to far
• he searching the formula in books or online
• dictionary-based pattern recognition

6/9

Human Approach

Rendered L

AT

EX: P (α,β)

n

(cos(aΘ)) The Naive Approach How does a reader understands the mathematical formula?

• he knows the symbols and structure,
• knowledge-based pattern recognition
• it was previously introduced in the paper (e.g. in definitions,

the text or in other referenced publications),

• analyse the context from near to far
• he searching the formula in books or online
• dictionary-based pattern recognition

6/9

Human Approach

Rendered L

AT

EX: P (α,β)

n

(cos(aΘ)) The Naive Approach How does a reader understands the mathematical formula?

• he knows the symbols and structure,
• knowledge-based pattern recognition
• it was previously introduced in the paper (e.g. in definitions,

the text or in other referenced publications),

• analyse the context from near to far
• he searching the formula in books or online
• dictionary-based pattern recognition

6/9

Human Approach

Rendered L

AT

EX: P (α,β)

n

(cos(aΘ)) The Naive Approach How does a reader understands the mathematical formula?

• he knows the symbols and structure,
• knowledge-based pattern recognition
• it was previously introduced in the paper (e.g. in definitions,

the text or in other referenced publications),

• analyse the context from near to far
• he searching the formula in books or online
• dictionary-based pattern recognition

6/9

Human Approach

Rendered L

AT

EX: P (α,β)

n

(cos(aΘ)) The Naive Approach How does a reader understands the mathematical formula?

• he knows the symbols and structure,
• knowledge-based pattern recognition
• it was previously introduced in the paper (e.g. in definitions,

the text or in other referenced publications),

• analyse the context from near to far
• he searching the formula in books or online
• dictionary-based pattern recognition

6/9

Adopt Human Approach

Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Adopt Human Behavior Let’s try to adopt the previous steps

• pattern recognition
• narrow down possible meanings from the structure of the

expression

• context analysis
• Near-Field-Analysis (NFA), e.g., extract identifier-definien

pairs from text, analyze definition environments, ...

• Far-Field-Analysis (FFA), e.g., overall topic of the paper,

citations, author’s field of interest, ...

7/9

Adopt Human Approach

Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Adopt Human Behavior Let’s try to adopt the previous steps

• pattern recognition
• narrow down possible meanings from the structure of the

expression

• context analysis
• Near-Field-Analysis (NFA), e.g., extract identifier-definien

pairs from text, analyze definition environments, ...

• Far-Field-Analysis (FFA), e.g., overall topic of the paper,

citations, author’s field of interest, ...

7/9

Adopt Human Approach

Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Adopt Human Behavior Let’s try to adopt the previous steps

• pattern recognition
• narrow down possible meanings from the structure of the

expression

• context analysis
• Near-Field-Analysis (NFA), e.g., extract identifier-definien

pairs from text, analyze definition environments, ...

• Far-Field-Analysis (FFA), e.g., overall topic of the paper,

citations, author’s field of interest, ...

7/9

Adopt Human Approach

Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Adopt Human Behavior Let’s try to adopt the previous steps

• pattern recognition
• narrow down possible meanings from the structure of the

expression

• context analysis
• Near-Field-Analysis (NFA), e.g., extract identifier-definien

pairs from text, analyze definition environments, ...

• Far-Field-Analysis (FFA), e.g., overall topic of the paper,

citations, author’s field of interest, ...

7/9

Adopt Human Approach

Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Adopt Human Behavior Let’s try to adopt the previous steps

• pattern recognition
• narrow down possible meanings from the structure of the

expression

• context analysis
• Near-Field-Analysis (NFA), e.g., extract identifier-definien

pairs from text, analyze definition environments, ...

• Far-Field-Analysis (FFA), e.g., overall topic of the paper,

citations, author’s field of interest, ...

7/9

Adopt Human Approach

Generic L

AT

EX: P_n^{(\alpha,\beta)}(\cos(a\Theta)) Adopt Human Behavior Let’s try to adopt the previous steps

• pattern recognition
• narrow down possible meanings from the structure of the

expression

• context analysis
• Near-Field-Analysis (NFA), e.g., extract identifier-definien

pairs from text, analyze definition environments, ...

• Far-Field-Analysis (FFA), e.g., overall topic of the paper,

citations, author’s field of interest, ...

7/9

Previous Research Projects

MathMLBen We developed a MathML gold standard and performed evaluations

• f nowadays L

AT

EX to MathML conversion tools.

mathmlben.wmflabs.org

Results: Many tools has trouble to understand formulae. MathMLTools We developed an open Java API for convenient MathML handling and easy access to useful services.

www.github.com/ag-gipp/MathMLTools

Evaluating Translations

www.dlmf.org

We performed automatic evaluation techniques on extracted formulae from the Digital Library of Mathematical Functions (DLMF) with the Computer Algebra System Maple. Results: We discovered two errors in the DLMF and one bug in Maple.

8/9

Previous Research Projects

MathMLBen We developed a MathML gold standard and performed evaluations

• f nowadays L

AT

EX to MathML conversion tools.

mathmlben.wmflabs.org

Results: Many tools has trouble to understand formulae. MathMLTools We developed an open Java API for convenient MathML handling and easy access to useful services.

www.github.com/ag-gipp/MathMLTools

Evaluating Translations

www.dlmf.org

We performed automatic evaluation techniques on extracted formulae from the Digital Library of Mathematical Functions (DLMF) with the Computer Algebra System Maple. Results: We discovered two errors in the DLMF and one bug in Maple.

8/9

Previous Research Projects

MathMLBen We developed a MathML gold standard and performed evaluations

• f nowadays L

AT

EX to MathML conversion tools.

mathmlben.wmflabs.org

Results: Many tools has trouble to understand formulae. MathMLTools We developed an open Java API for convenient MathML handling and easy access to useful services.

www.github.com/ag-gipp/MathMLTools

Evaluating Translations

www.dlmf.org

We performed automatic evaluation techniques on extracted formulae from the Digital Library of Mathematical Functions (DLMF) with the Computer Algebra System Maple. Results: We discovered two errors in the DLMF and one bug in Maple.

8/9

Previous Research Projects

MathMLBen We developed a MathML gold standard and performed evaluations

• f nowadays L

AT

EX to MathML conversion tools.

mathmlben.wmflabs.org

Results: Many tools has trouble to understand formulae. MathMLTools We developed an open Java API for convenient MathML handling and easy access to useful services.

www.github.com/ag-gipp/MathMLTools

Evaluating Translations

www.dlmf.org

We performed automatic evaluation techniques on extracted formulae from the Digital Library of Mathematical Functions (DLMF) with the Computer Algebra System Maple. Results: We discovered two errors in the DLMF and one bug in Maple.

8/9

Previous Research Projects

MathMLBen We developed a MathML gold standard and performed evaluations

• f nowadays L

AT

EX to MathML conversion tools.

mathmlben.wmflabs.org

Results: Many tools has trouble to understand formulae. MathMLTools We developed an open Java API for convenient MathML handling and easy access to useful services.

www.github.com/ag-gipp/MathMLTools

Evaluating Translations

www.dlmf.org

We performed automatic evaluation techniques on extracted formulae from the Digital Library of Mathematical Functions (DLMF) with the Computer Algebra System Maple. Results: We discovered two errors in the DLMF and one bug in Maple.

8/9

Previous Research Projects

MathMLBen We developed a MathML gold standard and performed evaluations

• f nowadays L

AT

EX to MathML conversion tools.

mathmlben.wmflabs.org

Results: Many tools has trouble to understand formulae. MathMLTools We developed an open Java API for convenient MathML handling and easy access to useful services.

www.github.com/ag-gipp/MathMLTools

Evaluating Translations

www.dlmf.org

We performed automatic evaluation techniques on extracted formulae from the Digital Library of Mathematical Functions (DLMF) with the Computer Algebra System Maple. Results: We discovered two errors in the DLMF and one bug in Maple.

8/9

Upcoming Projects

A real-time recommender system for semantic version of mathematical input included in the editor of Wikipedia articles.

• real-time recommendations
• ordered from most likely to impossible
• consider the context

9/9

Thank you for your attention!